Let F((.)) be a cumulative distribution function concentrated on (0,(INFIN)). Let N(t); t (GREATERTHEQ) 0 and U(t) be the associated renewal process;and renewal function, respectively. It is assumed that either F((.)) has a regularly varying tail with exponent -(alpha), 0 (LESSTHEQ) (alpha) (LESSTHEQ) 1, or;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);is slowly varying at infinity;Strong renewal theorems of Feller-Smith type are presented. These give the asymptotic behavior of the convolution (U*Q)(t) as t (--->) (INFIN) under the assumption that Q((.)) be regularly varying at infinity with exponent (beta) > -1;Weak and strong renewal theorems are given for generalized renewal functions G((.)) of the form;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);where a((.)) is regularly varying at infinity wtih exponent (beta) (GREATERTHEQ) -1;Second order estimates for U((.)) are presented for a class of distributions. These are possible through the use of a strong renewal theorem of Feller-Smith type;Limit theorems for N(t) are discussed and the (alpha) = 1 case is examined in some detail. Applications, which include the extreme value theory of regenerative stochastic processes, are given.